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Ginzburg-Landau Functionals and Reac...

Zhang, Edith Jin,

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Ginzburg-Landau Functionals and Reaction-Diffusion Equations in the Large-Graph Limit /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Ginzburg-Landau Functionals and Reaction-Diffusion Equations in the Large-Graph Limit // Edith Jin Zhang.
作者:	Zhang, Edith Jin,
面頁冊數:	1 electronic resource (109 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-11, Section: B.
Contained By:	Dissertations Abstracts International86-11B.
標題:	Applied mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31939348
ISBN:	9798314868492

Ginzburg-Landau Functionals and Reaction-Diffusion Equations in the Large-Graph Limit /
Zhang, Edith Jin,

Ginzburg-Landau Functionals and Reaction-Diffusion Equations in the Large-Graph Limit /Edith Jin Zhang. - 1 electronic resource (109 pages)

Source: Dissertations Abstracts International, Volume: 86-11, Section: B.

Graphons are a notion of infinite graphs that are spatially continuous rather than discrete (as are the nodes of a graph). They have emerged as a mathematical formalism that attempts to preserve the essential properties of graphs as their sizes approach infinity. In this dissertation, we consider two problems on graphs, and we let the size of the graphs approach infinity. We define graphon limits of these two graph problems, and we show that the limiting problems on graphons are consistent with their graph counterparts in the sense that the solutions of the graph problems converge to the solutions of the limiting graphon problems.In Chapters 3-6, we consider the problem of minimizing graph Ginzburg-Landau (GL) functionals. Graph GL functionals are relaxations of graph-cut functionals on graphs, and they have yielded a variety of insights in image segmentation and graph clustering. For sequences of graphs \uD835\uDC4A\uD835\uDC5B that converge to a limiting graphon \uD835\uDC4A, we show that the graph GL functional Γ-converges to a continuous and nonlocal functional that we call the graphon GL functional. We also show that graphon GL functionals Γ-converge, in a sharp-interface limit, to a graphon total variation (TV) functional. Using the Γ-convergence results, we conclude that the minimizers of the Γ-converging sequences of functionals converge to the minimizers of their limiting functionals. To obtain our convergence results, we require an extension of the underlying function spaces; we discuss the need for a probabilistic interpretation (via Young measures) of the variational problems in the graphon limit.In Chapters 7-8, we study graph RD equations, which are certain systems of differential equations that are defined on the nodes of a graph. Consider a sequence of graphs that converges to a limiting graphon. We show that the solutions of the sequence of graph RD equations converge to the solution of a limiting graphon RD equation, which we define to be a continuous RD equation that has a nonlocal diffusion term. Furthermore, we show that a sequence of stochastic particle processes (that consist of a random walk and a birth-death process) on the sequence of graphs converges to the solution of the graphon RD equation.The graphon limits of graph problems can be viewed as continuous and nonlocal counterparts of discrete graph problems, which tend to be large systems of coupled equations. Our results establish precedent and intuition for future work on graphon limits of graph problems, and they build the connection between graphon theory and nonlocal analysis.

English

ISBN: 9798314868492Subjects--Topical Terms:

2122814
Applied mathematics.
Subjects--Index Terms:

Nonlocal analysis

Ginzburg-Landau Functionals and Reaction-Diffusion Equations in the Large-Graph Limit /
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500 $a Advisors: Du, Qiang.
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520 $a Graphons are a notion of infinite graphs that are spatially continuous rather than discrete (as are the nodes of a graph). They have emerged as a mathematical formalism that attempts to preserve the essential properties of graphs as their sizes approach infinity. In this dissertation, we consider two problems on graphs, and we let the size of the graphs approach infinity. We define graphon limits of these two graph problems, and we show that the limiting problems on graphons are consistent with their graph counterparts in the sense that the solutions of the graph problems converge to the solutions of the limiting graphon problems.In Chapters 3-6, we consider the problem of minimizing graph Ginzburg-Landau (GL) functionals. Graph GL functionals are relaxations of graph-cut functionals on graphs, and they have yielded a variety of insights in image segmentation and graph clustering. For sequences of graphs \uD835\uDC4A\uD835\uDC5B that converge to a limiting graphon \uD835\uDC4A, we show that the graph GL functional Γ-converges to a continuous and nonlocal functional that we call the graphon GL functional. We also show that graphon GL functionals Γ-converge, in a sharp-interface limit, to a graphon total variation (TV) functional. Using the Γ-convergence results, we conclude that the minimizers of the Γ-converging sequences of functionals converge to the minimizers of their limiting functionals. To obtain our convergence results, we require an extension of the underlying function spaces; we discuss the need for a probabilistic interpretation (via Young measures) of the variational problems in the graphon limit.In Chapters 7-8, we study graph RD equations, which are certain systems of differential equations that are defined on the nodes of a graph. Consider a sequence of graphs that converges to a limiting graphon. We show that the solutions of the sequence of graph RD equations converge to the solution of a limiting graphon RD equation, which we define to be a continuous RD equation that has a nonlocal diffusion term. Furthermore, we show that a sequence of stochastic particle processes (that consist of a random walk and a birth-death process) on the sequence of graphs converges to the solution of the graphon RD equation.The graphon limits of graph problems can be viewed as continuous and nonlocal counterparts of discrete graph problems, which tend to be large systems of coupled equations. Our results establish precedent and intuition for future work on graphon limits of graph problems, and they build the connection between graphon theory and nonlocal analysis.
546 $a English
590 $a School code: 0054
650 4 $a Applied mathematics. $3 2122814
650 4 $a Mathematics. $3 515831
650 4 $a Statistics. $3 517247
650 4 $a Theoretical physics. $3 2144760
653 $a Nonlocal analysis
653 $a Calculus of Variations
653 $a Graph theory
653 $a Graphons
653 $a Graphon GL functional
690 $a 0364
690 $a 0405
690 $a 0753
690 $a 0463
710 2 $a Columbia University. $b Applied Mathematics. $e degree granting institution. $3 3823064
720 1 $a Du, Qiang $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-11B.
790 $a 0054
791 $a Ph.D.
792 $a 2025
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31939348